To verify whether the birefringence corresponds to the polymer chain morphology, we introduce the Weissenberg number \(Wi\)  as an indicator.
It is known that for a polymer coil to stretch, \(Wi\)  must at least exceed a critical value of \(\frac{1}{2}\)  [1]}, [2]}, [3]}.
In this study, we calculate \(Wi\)  as the product of the strain rate \(\dot{\epsilon }_\mathrm {mid}\)  and the relaxation time \(\lambda _{E}\) . The latter is constant [2]}, and so
the curve of \(Wi\)  has the same shape as that of the strain rate.
Figure REF  show that \(Wi\)  remains larger than the critical value of \(\frac{1}{2}\)  until the pinching off of the liquid polymer filament, while the strain rate is constant within the EC regime.
Moreover, it is clear that each plot of \(Wi\)  has two turning points: a maximum followed by a minimum.
The normalized times at the first turning point are 0.86 and 0.57 for the PEOs with molecular weight \(M_{w} = 1\times 10^6\)  and \(5\times 10^6\) , respectively, both of which are within the IC region.
For both PEOs, the second turning point of \(Wi\) , which is at a value greater than \(\frac{1}{2}\) , is located at the transition from the IC to the EC regime.
The existence of two turning points, which cause the plot of \(Wi\)  to have a mountainous shape from the IC to the EC regime, has also been reported for other polymer solutions [1]}, [3]}.
From a comparison of the experimental results for the birefringence and \(Wi\) , we can confirm that the birefringence becomes constant at the point of transition from the IC to the EC regime, where the \(Wi\)  data show that the polymer chain gradually becomes stretched from the coiled state.
Therefore, we argue that the measured birefringence can be used as an indicator to investigate the coil-stretch transition in our rheo-optical technique that uses both a high-speed polarization camera and a CaBER-DoS system.
